MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION

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1 (ISO/IEC Certiied) Page No: /6 WINTER 5 EXAMINATION MODEL ANSWER Subject: ENGINEERING MATHEMATICS (EMS) Subject Code: 76 Important Instructions to eaminers: The model answer shall be the complete solution or each and ever question on the question paper. Numerical shall be completel solved in a step b step manner along with step marking. All alternative solutions shall be oered b the epert along with sel-eplanator comments rom the epert. In case o theoretical answers, the epert has to write the most acceptable answer and oer comments regarding marking scheme to the assessors. In should oer the most convincing igures / sketches / circuit diagrams / block diagrams / low diagrams and oer comments or step marking to the assessors. In case o an missing data, the epert shall oer possible assumptions / options and the ensuing solutions along with comments to the assessors or eective assessment. In case o questions which are out o the scope o curricular requirement, the epert eaminer shall solve the question and mention the marking scheme in the model answer. However, the eperts are requested to submit their clear cut opinion about the scope o such question in the paper separatel to the coordinator. Eperts shall cross check the DTP o the inal drat o the model answer prepared b them.

2 (ISO/IEC Certiied) Subject Code: (76) Winter 5 Page No: /6 Model wers ) Attempt an TEN o the ollowing: (a). I z z i, e v a lu a te z z z i i 6 i 9 6 i i (b) E p ress + i in m o d u lu s an d am p litu d e o rm. L e t z i, r z a n d, ta n ta n ta n z r c o s i s in i c o s i s in c). I 6 lo g, in d 6 lo g, 6 lo g log lo g lo g lo g lo g

3 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model answers. d) D eine even and odd unction. E v e n u n c tio n :- I is a n e v e n u n c tio n, th e n th e u n c tio n O d d u n c tio n :- I, th e n th e u n c tio n is a n o d d u n c tio n (e) E v a lu a te : () E v alu ate: sin tan 5. s in ta n 5 s in ta n 5 s in. ta n

4 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers. g) E v a lu a te : s in s in s in s in s in.. s in lo g lo g 9 lo g 9 lo g 8 lo g (h) I e co s, in d e c o s d d e sin c o s e. + d e sin c o s (i) I lo g sin, in d d

5 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 5/6 Model wers ) lo g s in d d d c o s O R s in d s in d s in d + d c o s d c o s d O R d s in d s in d d c o t d O R c o t d d j) F in d, i sin, co s 6 d s in, c o s 6 d c o s a n d d s in 6 d d + d d d d s in 6 c o s d d. s in c o s c o s sin k) S h o w th a t th e ro o t o 9 lie s b e tw e e n a n d. L e t 9 9 ro o t lie s b e tw e e n a n d l) F in d th e irs t ite ra tio n b u s in g J a c o b i 's m e th o d o r th e o llo w in g s ste m o e q u a tio n s : 5 z, z 5, 5 z In itia l a p p ro im a tio n s : z 5 z 5 z z 5.,.7 5, z

6 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 6/6. Attempt an FOUR o the ollowing: 6 a) Find cube root o unit and show that one root is square o the other. w w P u t w z z i, r z ta n G e n e ra l p o la r o rm is, z r c o s n i s in n w c o s n i s in n w c o s n i s in n n n w c o s i s in ; n,, w h e n n w c o s i s in w h e n n w c o s s in i i w h e n n w c o s i s in i c o n s id e r i i w w i b) c o s i s in c o s is in S im p li: u s in g D e -M o iv e r's th e o re m c o s i s in c o s 5 is in 5

7 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 7/6 Model wers ) c o s i s in c o s i s in c o s i s in c o s 5 is in 5 c o s i s in c o s is in 5 c o s i s in c o s i s in ++ + c o s i sin c o s i sin 5 c o s i sin c) I s in A ib i p ro v e th a t: i) c o s h B s in h B ii) s in A c o s A s in A ib i s in A c o s ib c o s A s in ib i s in A c o s h B i c o s A s in h B i s in A c o s h B, c o s A s in h B i) s in A c o s h B c o s A s in h B c o s h B s in h B c o s h B s in h B sin A cos A ii) s in A c o s h B c o s A s in h B s in A c o s A s in A c o s A cosh B sinh B d) a I lo g s h o w th a t a a lo g a a a a a lo g lo g a a + a a log. a a a log a

8 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 8/6 Model answers ) e) U s in g E u le r's e p o n e n tia l o rm u la p ro v e th a t: s in c o s s in c o s i i i i e e e e i i i i i i e e e e i i i i i i i i e e e e e e e e i i e e e ) I s h o w th a t L e t c o n s id e r o 7 7 o

9 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 9/6 Model wers ) Attempt an FOUR o the ollowing: 6 a) 5 I an d t sh o w th at t 5 t t t b) I t 5 s in t., th e n s h o w th a t t t t 5 s in t. =5 sin t. =5 sin t. = t c) E v a lu a te :. =

10 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers ) d) s in s in E v a lu a te : s in c o s s in s in c o s = s in s in s in s in s in s in e) c o s c o s E v a lu a te : c o s c o s c o s c o s P u t h, a s, h h c o s h h h s in h h

11 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers ) sin h h h ) lo g E v a lu a te : 9 P u t h, a s, h lo g h h h 9 h lo g h 9 6 h h 9 lo g h h h 6 h lo g h h 6 h h lo g e

12 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers ) Attempt an FOUR o the ollowing: 6 a) I u a n d v a re d ie re n tia b le u n c tio n s o a n d u, w h e re v v d u d v v u d d th e n p ro v e th a t d v L e t u, v, a re s m a ll in c re m e n ts in u, v, re s p e c tiv e l c o rre s p o n d in g to in c re m e n t in. u u v v u u v v u u u v v v u v v u u ( v v ) v v v v u u v v v v v u u v v v v v v v v u u v u v v u v d u d v v u d d d a s, v v b) B u s in g irs t p rin c ip le in d th e d e riv a tiv e o h d h h c o s d h h c o s h c o s

13 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers ) d h h h s in s in h d h h s in s in h h h s in d h s in. d h h h d d s in sin c) I s in, in d d P u t ta n ta n s in ta n s in s in s e c c o s s in ta n s in d d

14 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers ) OR I s in, in d d P u t c o t c o t s in c o t s in s in c o t c o s ec s in d d) F in d i d G iv e n, c o s c o s c o s lo g lo g lo g lo g c o s lo g lo g lo g c o s lo g s in lo g c o s d c o s d ta n lo g c o s d c o s ta n lo g c o s d

15 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 5/6 Model wers ) e) p q p q d I. s h o w th a t d p q lo g lo g p q p q lo g lo g p q lo g p lo g q lo g p q lo g d d p q p q d d p q d p q p q d d d q p q d p q p d q q p q d p q p p d q p d q p d d d ) I s in t s in t, c o s t c o s t, in d a t t d s in t s in t, c o s t c o s t c o s t 6 s in t c o s t dt d s in t 6 c o s t s in t dt d dt c o s t 6 s in t c o s t d d s in t 6 c o s t s in t d dt c o s t s in s in t c o s t t d c o s t c o s t c o t t d s in t c o s t a t t d c o t

16 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 6/6 Model wers 5) Attempt an FOUR o the ollowing: 6 a) 6 E v a lu a te : 6 lo g lo g b) E v a lu a te : lo g lo g P u t h a s, h lo g h lo g h h h lo g h h lo g h h h lo g h h h lo g h h h lo g e lo g e

17 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 7/6 Model wers 5) c) F in d th e a p p ro im a te ro o ts o th e e q u a tio n b b is e c tio n m e th o d. L e t ro o t lie s in, a b < the root lies in (.5,) b the root lies in (.75,) b OR L e t ro o t lie s in, a b a b d) S h o w th at ro o t o th e eq u atio n in, an d in d it b u sin g N ew to n -R ap h so n m eth o d p ero r m in g tw o iteratio n s.

18 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 8/6 Model wers 5) Let, ' ' 8 Initial root = ' ' OR Let, ' ' 8 Initial root n ' ' n n n n,,

19 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 9/6 Model answers 5) e) Solve the ollowing equations, Using Gauss eination method: z, z, z z z z z 6 z 5 z 8 and 9 6 z z 7 5 z 5 5 z 7 5 z 8 8 z Note: In the above solution, irst is einated and then z is einated to ind the value o irst. I in case the problem is solved b eination o another unknown i. e., either irst or z, appropriate marks to be given as per above scheme o marking. ) Solve the ollowing equations b Gauss-Seidel method: 5 9, 5 z, 5 z z 5 z 5 6 S ta rtin g w ith z z

20 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers 5).. z z ) Attempt an FOUR o the ollowing: 6 a) sin m d d I e p ro v e th a t m d d e m sin d e m sin m m d d d d m d d d d d d m d d d d d m. m b () d d d d m d d OR e m s in e d m s in m m e d m s in d d m s in m e m d d d d m d d

21 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers 6) d d m d d b) d I a ( s in ), a c o s in d a t = d a ( s in ), a c o s d a d asin d c o s d d d d d d a a s in c o s d d s in c o s c o s tan d d sec d d d sec d d d d d sec a co s d d sec sec a a co s

22 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers 6) a t d s e c d a a d d a c) Using Regula-Falsi method, ind the root o the equation. Let 5 the root lies in (,) a b b a (5 ) ( ). 6 7 b a < the root lies in (.67,) the root lies in (.5,) OR

23 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers 6) 5 the root lies in (,) a b a b a b b a b a d) Solve the ollowing equations b Jacobi s method. z 7, z 8, z 5 7 z 8 z z 5 S ta rtin g w ith z z z. z...

24 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: /6 Model wers 6) e) Solve the ollowing equations b Gauss eination method: z, z, 8 z z z 8 z z z 6 z and 6 6 z + 7 z z z 5 8 z 9 8 z z Note: In the above solution, irst is einated and then z is einated to ind the value o irst. I in case the problem is solved b eination o another unknown i. e., either irst or z, appropriate marks to be given as per above scheme o marking ) Using Newton-Raphson method to evaluate decimal places. Let, correct to three 7

25 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 5/6 Model wers 6) ' Initial root ' 7 = '.7 () ' ' OR Let, 7 ' Initial root n ' ' n n n

26 (ISO/IEC Certiied) Subject Code: (76) Winter-5 Page No: 6/6 Model wers 6) n,, Important Note In the solution o the question paper, wherever possible all the possible alternative methods o solution are given or the sake o convenience. Still student ma ollow a method other than the given herein. In such case, irst see whether the method alls within the scope o the curriculum, and then onl give appropriate marks in accordance with the scheme o marking